Can anyone please help me in equation form with the answers to the following questions?
1. Give the range of the function y = tan^-1 x
2. Give the value of tan^-1 5
There are two ways to do this problem. One way is the cowardly way they teach in school using a graph in that case look below to see that the range is,Originally Posted by Kiwigirl
$\displaystyle y\in (-\pi/2,\pi/2)$
The second way. The function $\displaystyle \tan^{-1}(x)$ is defined as the inverse function of the function,
$\displaystyle \tan x \mbox{ on } -\pi/2<x<\pi/2$. Now there is a rule that the range of the inverse function is the domain of the orignal function. Since the domain of the original function is $\displaystyle (-\pi/2,\pi/2)$ thus the range of $\displaystyle \tan^{-1}(x)$ is $\displaystyle (-\pi/2,\pi/2)$
There is actually a third way (much more advanced). Find all such $\displaystyle y$ such as the equation $\displaystyle y=\tan^{-1}(x)$ has a solution.